Simultaneous equations: Section 3
Example 6. Solve this system of three equations in three unknowns:
| 1) | x | + | y | − | z | = | 4 |
| 2) | x | − | 2y | + | 3z | = | −6 |
| 3) | 2x | + | 3y | + | z | = | 7 |
The strategy is to reduce this to two equations in two unknowns. Do this by eliminating one of the unknowns from two pairs of equations: either from equations 1) and 2), or 1) and 3), or 2) and 3).
For example, let us eliminate z. We will first eliminate it from equations 1) and 3) simply by adding them. We obtain:
4) 3x + 4y = 11
Next, we will eliminate z from equations 1) and 2). We will multiply equation 1) by 3. We call the resulting equation 1' ("1 prime") to show that we obtained it from equation 1):
| 1') | 3x | + | 3y | − | 3z | = | 12 |
| 2) | x | − | 2y | + | 3z | = | −6 |
| ______________________________________________________________________________________ | |||||||
| 5) | 4x | + | y | = | 6 | ||
We now solve equations 4) and 5) for x and y.
Let us eliminate y. We will multiply equation 5) by −4, and add it to equation 4):
| 5') | −16x | − | 4y | = | −24 |
| 4) | 3x | + | 4y | = | 11 |
| ______________________________________________________________________________________ | |||||
| −13x | = | −13 | |||
x = 1
To solve for y, let us substitute x = 1 in equation 4):
| 3 + 4y | = | 11 |
| 4y | = | 11 −3 |
| 4y | = | 8 |
| y | = | 2 |
Finally, to solve for z, substitute these values of x and y in one of the original equations; say equation 1):
| 1 + 2 − z | = | 4 |
| −z | = | 4 − 3 = 1 |
| z | = | −1 |
Problem 8. Solve this system of equations.
| 1) | x | + | y | + | z | = | 6 |
| 2) | x | − | y | + | z | = | 2 |
| 3) | x | + | 2y | − | z | = | 2 |
Eliminate y, for example, from equations 1) and 2, and then from equations 2) and 3).
Add equations 1) and 2):
4) 2x + 2z = 8
Next, multiply equation 2) by 2, and add it to 3):
| 2') | 2x | − | 2y | + | 2z | = | 4 |
| 3) | x | + | 2y | − | z | = | 2 |
| ______________________________________________________________________________________ | |||||||
| 5) | 3x | + | z; | = | 6 | ||
Solve 5) with 4). Multiply 5 by −2:
| 5') | −6x | − | 2z | = | −12 |
| 4) | 2x | + | 2z | = | 8 |
| ______________________________________________________________________________________ | |||||
| −4x | = | −4 | |||
| x | = | 1 | |||
To solve for z , substitute x = 1 in equation 5):
| 3 + z | = | 6 |
| z | = | 3 |
Finally, to solve for y, substitute these values of x and z in one of the original equations; say equation 1):
| 1 + y + 3 | = | 6 |
| y | = | 2 |
Always verify the solution by plugging the numbers into each of the three equations.
Problem 9. Solve this system of equations.
| 1) | x | + | y | − | z | = | 1 |
| 2) | 8x | + | 3y | − | 6z | = | 1 |
| 3) | −4x | − | y | + | 3z | = | 1 |
Here is the solution: x = 2, y = 3, z = 4.
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